Figure 8 – uploaded by Kathleen Howell

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Figure 8 (a) Selected L1 Lyapunov orbits within the Earth—Moon system. Orbit period: P & 11.95 days (blue), P & 13.26 days (orange), P & 16.81 days (red)

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Fig. 1 Schematics for the gravity force exerted by a particle on a rigid body. be obtained by replacing the vector p with the vector

Table 1 Coefficients in the characteristic equation for representative spacecraft configurations at an equilibrium point within the CR3BP. Useful parameters are defined at the bottom of this table. The quantities d and r denote distances from the primaries, yw is the mass parameter in the CR3BP, k; and kz are the inertia ratios for an asymmetric body as defined in Eq. (3), o is the spin rate observed within the CR3BP rotating frame, J, and J, describe the axial and transversal inertia for an axisymmetric spacecraft, respectively Combining all the conditions for in-plane and out- of-plane stability, stable configurations are identified on a modified Beletskii-DeBra—Delp diagram. At a collinear point, a diagram as in Fig. 2 is expected. Referring to this figure, there exist two regions of stable equilibria. The boundaries for region I are independent of the mass paremeter and the collinear point location information, and therefore, they do not vary with the planetary system. Moreover, region I identically appears Selecting a single equilibrium location within the CR3BP rotating frame and fixing the mass parameter, it is possible to evalua (14) as the spacecraft iner vary. Recording orienta ki —ka chart is essential Delp diagram, that is tion s body interaction. The Belets is originally developed to grap conditions for a body in circu body gravitational field 27,28 modi te stabili tia tabi fied kii— _ It ty conditions in Eq. ratios, k, and kg, ity information on a y creating a Beletskii-DeBra-— to render a three- DeBra—Delp diagram hically represent stability ar orbit within a central can be demonstrated that a condition b > 0 reduces to an inequality kjk < 0 at a collinear point and at an equilateral point [25]. The inequality k,ko <0 excludes the first and third quadrants of the Beletskii-DeBra—Delp diagram from stable configurations. The remaining conditions a > 0 and a? > 4b are more complex to solve. It is common equilibria. The boundaries for region I are independent

Fig. 2 Stability diagram for rigid body at the L3 point within the Earth—-Moon system. Colored areas indicate linearly stable configurations.

Fig. 3 Size variation for linearly stable regions II and IV as a function of the mass parameter. Red line marks the area value for a rigid body in circular orbit about a single attractor. Area values are presented as percentage over the total area of the stability diagram.

Fig. 4 Stability diagram for rigid body at the L4 point within the Earth—Moon system. Colored areas indicate linearly stable configurations.

(c) Size variation for linearly stable regions as a function of the mass parameter (b) Three-body problem at L1 within the Earth- Moon system Fig. 5 Stability diagram for a spinning axisymmetric satellite. In subfigures (a) and (b), colored regions indicate stable configurations, and red, blue, and black contours denote respectively a = 0, b = 0, and a? = 4b. In subfigure (c), a dashed line marks the value for the linear stability area in the two-body problem. Area values are presented as percentage over the total visible area of the stability diagram.

Fig. 6 Orbit—attitude resonance conditions within the planar center subspace (i.e., small planar periodic orbits) for the collinear points.

(b) Angular velocity as measured by an inertial observer. Horizontal line marks an angular velocity equal to one

(c) Angular velocity average over half orbit period as a function of the perilune radius, as measured by an inertial observer. Horizontal line marks an average angular velocity equal to one

(a) Selected L1 Lyapunov orbits within the Earth-Moon system. Orbit period: P © 11.95 days (blue), P © 13.26 days (orange), P * 16.81 days (red) Fig. 8 Sample evolution of gravity gradient profile for a spacecraft along selected non-linear planar orbits in vicinity of a collinear point. Spacecraft principal axes of inertia are artificially aligned with the rotating frame at all times.

(b) Angular velocity components as measured with respect to an inertial observer

(b) Stability map as a function of the reference orbit and inertia characteristics. Reference orbits are selected from an L1 Lyapunov family and identified by the corresponding period. Satellite inertia is described by the ratio k = I,/I; Fig. 10 A stability map for a disk-like (I, > [,) axisymmetric spinning satellite on planar orbit. Motion is within the Earth— Moon system.

Fig. 11 Appearance of chaos in planar pitch motion for increasingly large L1 Lyapunov reference orbits in the Earth— Moon system. Spacecraft may be described as a rod-like satellite with transversal inertia twice as large as the axial inertia. constructing a new map: given Fig. Il as an example,

(b) Maximum pitch angle along one revolution of the reference orbit as a function of the orbit amplitude, A,, and inertia ratio kg Fig. 12 Grid search map representing planar pitch rotations for a satellite in L1 Lyapunov orbit within the Earth—Moon system.

(b) Family of attitude periodic solutions for different values of inertia ratio, k Fig. 13 An example of periodic solution structure on a stability map for a disk-like axisymmentric spacecraft.

(a) Poincaré map. Red arrow points at the center of an island. where there exists a periodic solution

Fig. 14 An example of structure that is associated to attitude periodic solutions on a Poincaré map.

(a) Grid searh map. Red dots mark attitude periodic solutions. Arrow indicates first computed member of the family Fig. 15 An example of structure that is associated to attitude periodic solutions on a grid search map.

solutions is computed and overlaid on the grid search map. First, identify an initial periodic solution, e.g., for kj = —0.8, that is located within the region of interest and one that is consistent with the grid search definition. Then, step onto a nearby motion by a small modification of the mass distribution, and re-converge the solution to be precisely periodic via, for instance, differential corrections. Since the solution combines orbit and attitude periodic dynamics, converging on a new periodic motion also requires adjustments for the reference orbit. Reiterating this process, which is a simple implementation of a continuation algorithm, a family of attitude periodic solutions across different spacecraft mass configurations is produced. The family is represented by the corresponding inertia ratio, ks, and the reference trajectories are identified by their amplitude, Ay. The family of attitude behaviors that is generated for different values of the inertia ratio k3 can, therefore, be overlaid on the grid search map by plotting the orbit amplitude, A,, for representative members, as in Fig. 15(a). The family of attitude periodic responses in Fig. 15(a). The family of attitude periodic responses

Fig. 16 Identification of attitude periodic solutions along northern L1 halo orbits in the Earth—Moon system.

(a) Grid search map for the nutation angle. Arrow points to a region of bounded motion

reference path. Recall that the Euler angles are all zero Euler sequence map is displayed in Fig. 16(a) as an example. The resulting representation supports the identification of a pair of northern halo orbit, denoted by the orbit amplitude in the Z direction, A,, and inertia ratio, k;, that enables a vehicle to maintain an approximate alignment of the body axes with the CR3BP rotating frame over one revolution along the reference path. Recall that the Euler angles are all zero

Fig. 17 Schematics of two branches for a higher dimension family intersecting in a parametric space. A first extension to the CR3BP model, important for several celestial systems, is the inclusion of an Comparison of torque-free solutions with attitude periodic motions within the CR3BP supports the hypothesis that a certain type of periodic motion is fundamentally due to the existence of a three-body interaction [40,48]. If this type of dynamical structure is representative of motion in a higher-fidelity model, however, the impact is largely unexplored, at the present time.

than adjusting the natural attitude dynamics after the trajectory design is completed by the addition of a time-varying control torque throughout each trajectory arc, design a sequence of orientations that already balances the natural rotational behavior, while steering a spacecraft along a desired path. Following an instantaneous and artificial modification of the pointing direction, a spacecraft rotates consistent with the natural flow and does not, in general, maintain a fixed direction relative to any selected reference (as it does within a decoupled TnH guidance). Figure 18 presents converged trajectories for a disk-like sail, one that is initially displaced, in terms of both position and velocity, from the nominal orbit and its goal is to return to the reference within one revolution of the orbit. A sail angular momentum component equal to 10 nondimensional units along the sail normal direction is assumed. An initial guess may be supplied by a solution from a decoupled TnH algorithm. The attitude Fig. 18 ‘Trajectory for a coupled orbit—attitude TnH guidance, one that is applied to modified L1 halo orbits in the Sun-— Earth system. Initial perturbation is approximately 75000 km in position and 3 m/s in velocity. Sail is modelled as an axisymmetric vehicle with inertia ratio k = J,/I, = 2, and lightness parameter 6B = 0.035. rbit. A sail angular momentum component equal to

Fig. 19 Inertial spacecraft orientation via the quaternion vector for a coupled orbit—attitude TnH guidance, one that is applied to modified L1 halo orbits in the Sun—Earth system. Sail is modelled as an axisymmetric vehicle with inertia ratio k = I,/I, = 2, and lightness parameter 8 = 0.035. history appears in Fig. 19, as it is described by a quaternion vector relative to the inertial frame. That profile comprises both reorientation events, as well as the subsequent natural evolution for the attitude configuration. Ideally (i.e., neglecting any perturbation to the selected dynamical model and uncertainties in the state variables), trajectories in Fig. 18 do not require any additional control action after a sail realignment. Further details on the construction of this example are available in Guzzetti [43]. In general, this example of spinning solar sail, demonstrates the possibility to directly leverage the natural attitude dynamics information within a series of orientation configurations that guides a spacecraft along a desired path. This approach may be a good compromise between the numerical robustness of a decoupled TnH guidance and the simplification of an attitude control scheme, one that is possible via a more accurate coupled dynamics description in the early stages of the design process. description in the early stages of the design process. 1umerical robustness of a decoupled TnH guidance and

Figure 8 – uploaded by Kathleen Howell

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